Physical properties of ground states on static spacetimes of compact objects

In the talk I will mainly deal with a scalar quantum field propagating on given, static spacetimes. In is known, that for static spacetimes with horizons, such as the Schwarzschild spacetime, the ground states exhibit intriguing behavior near the horizon. For instance, the expectation value of the energy density diverges to minus infinity at the Schwarzschild horizon. On the other hand, it is also known that for static spacetimes without horizons no such divergency should be expected (the ground states are Hadamard states). However, it is not known, whether the ground states for spacetimes which only merely avoid collapse do exhibit large negative energy densities anywhere. In my talk I will show that Quantum Weak Energy Inequalities can be employed to yield an answer to this question. This general method, which avoids cumbersome direct calculations, provides physically tight bounds if the spacetimes are isometric to Schwarzschild up to a small distance from the horizon. I will discuss some recent results of mine, together with the problems currently under attack and some possible future generalizations.